Complex fluids are fluids that are homogeneous at macroscopic scales and disordered at microscopic scales, but possess structures on a mesoscopic scale. Polymeric liquids, bio-fluids, bio-materials, and nanocomposites all come under the category of complex fluids. From a mathematician's perspective, we conduct research in the areas of differential equations, applied analysis, dynamical systems, probability, stochastic differential equations, nuerical computation, continuum mechaincs theory, kinetic theory, statistical mechanics, quantum mechanics and condensed matter physics.

My work includes development of theories for complex fluids, fluid dynamic and rheological study of the fluids, and computating polymeric flow. More specifically, mesoscopic domain theory and kinetic theory for biaxial liquid crystal polymers and suspension flows in general are developed to study their phase,rheology and material properties. Wave propagation in the biaxial liquid crystal polymers subject to active electric and/or magnetic field is also our research object.

In certain regions of the material parameter space (γ, λ) and at concentration (N=5.5,6.28,7.84), we have found that the mesoscale major director $$m₁ changes its alignment from the steady LR (Log-rolling) to the steady FA (Flow Alignment), via a couple of exotic out-of-plane time periodic motions: (1) Mixed Kayaking: MK and (2) Fluttering Kayaking: FK at intermediate Peclet numbers.

Mixed Kayaking: is a comined motion of the tilted kayaking of $$m₁ and the full kayaking motion of the secondary director $$n₁. The eigen vectors $$m₁ rotates about a tilted axis while $$m₂, the second eigenvector of the second moment (M), collectively go through a coordinated tilted motion about their respective tilted axis. On the other hand, $$n₁ and $$n₂ (the second eigenvector of the second moment N) rotate about the vorticity axis in a weakly non-planar fashion, imitating a full-blown kayaking motion against the flow flow-gradient plane.

Fluttering Kayaking: is a motion where $$m₁ wags in the flow flow-gradient plane while $$m₂ goes through a truly nonplanar circular motion. The coordinated motion of $$m₁ and $$m₂ is reminiscent of the fluttering fall of a leaf or a feather in the air. In the meantime, $$n₁ and $$n₂ rotate around two tilted axes coordinately, analogous to the major director motion of the tilted kayaking.

In both the MK and the FK motion, the orbits of the pair of directors going through the tilted kayaking osculate each other near the vorticity axis. The eigen-frames of M and N is skewed most of the time in the time-periodic motion and the angle between $$m₁ and $$n₁ oscillates between $40o$ and $140o$ within a period. The degrees of orientation about the directors fluctuate as well.